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受欢迎的微积分 >

integral of 6/(sqrt(x^2-4))

解答

\int \frac{6}{x ^{2} - 4} d x

解答

6 ln (\frac{1}{2} x^{2} - 4 + x) + C

求解步骤

\int \frac{6}{x ^{2} - 4} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int \frac{1}{x ^{2} - 4} d x

使用三角换元法

= 6 \cdot \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 6 ln ∣ tan (u) + sec (u) ∣

u = arcsec (\frac{1}{2} x)

代回

= 6 ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x))

化简

6 ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x)) : 6 ln (\frac{1}{2} x^{2} - 4 + x)

= 6 ln (\frac{1}{2} x^{2} - 4 + x)

解答补常数

= 6 ln (\frac{1}{2} x^{2} - 4 + x) + C

作图

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